## 1. Introduction

[2] Radio occultation soundings of the Earth atmosphere allows for observations of temperature, pressure, and humidity. In the radio occultation (RO) technique these geophysical parameters are related to the Doppler shift imposed by the atmosphere on a signal emitted by a GPS satellite and received by a low Earth orbiting (LEO) satellite [see, e.g., *Kursinski et al.*, 1997, 2000; *Rocken et al.*, 1997]. During a RO observation the LEO satellite sets behind the Earth limb and thus provides a vertical scan of the atmosphere. Because of refractivity index gradients in the atmosphere, radio waves are bent as the radio signal traverses the atmosphere. The bending angles of the rays are directly related to the measured Doppler shifts through the occultation geometry and may be inverted into a profile of refractivity using the Abel transform [*Fjeldbo et al.*, 1971]. As GPS signals received by a LEO satellite propagates through the ionosphere the measured Doppler shifts and bending angles include unwanted contributions from the ionosphere. For reconstruction of the refractivity profile of the neutral atmosphere this effect must be removed. At GPS frequencies the neutral atmosphere is nondispersive, whereas the ionosphere is dispersive. This allows for removal of the ionospheric contributions to first order through a linear combination of pairs, corresponding to the two frequencies in the GPS system, of either signal phase [*Spilker*, 1980], phase modulation [*Melbourne et al.*, 1994], bending angle [*Vorob'ev and Krasil'nikova*, 1994], or Doppler shift [*Ladreiter and Kirchengast*, 1996]. These techniques are referred to as ionospheric calibration.

[3] Even when ionospheric calibration is applied, the retrieved bending angle profiles are dominated by noise from residual errors in the calibration at heights above 40–60 km. When refractivity profiles are computed from bending angle profiles through the Abel transform, errors propagate from higher altitudes to lower altitudes. Therefore it is essential to reduce these errors to get accurate refractivity observations in the stratosphere. A common approach to reduce high-altitude bending angle errors is to use statistical optimization (SO) where observed bending angle profiles are combined with a priori or first-guess profiles. First-guess profiles are normally obtained from some climate model, though, in principle, any kind of a priori information about the bending angle profile can be used as the first guess, for example, a bending angle profile obtained from a weather prediction model. In this study, only first-guess profiles obtained from a climate model are considered.

[4] If the error covariances, **B**_{obs}, of the measured bending angle profiles and the error covariance of the first-guess profile, **B**^{guess}, are known, the statistical optimal bending angle profile, α_{opt}, is found by minimizing the cost function

where α_{obs} and α_{guess} are the observation and the first-guess bending angle vectors, respectively [see, e.g., *Gorbunov et al.*, 1996; *Healy*, 2001]. The solution that minimizes (1) is

with corresponding error covariance *B*_{opt} [*Rodgers*, 2000]:

In practice, however, the error covariances are not known. While the observation error covariances may be estimated from the topmost points in the observed bending profiles (where the observation noise overshadows the signal from the neutral atmosphere), only rough estimates are available for the error covariances of the climate models which are normally used as the first guess in SO. The lack of knowledge of climate model error covariances constitutes the main limitation to the use of SO for RO data. Hence, for practical application of SO to RO data, some assumptions must be made about these error covariances.

[5] A common approach, which has been applied in a number of studies [*Gorbunov et al.*, 1996; *Sokolovskiy and Hunt*, 1996; *Hocke*, 1997; *Rocken et al.*, 1997; *Gorbunov and Gurvich*, 1998; *Steiner et al.*, 1999; *Hajj et al.*, 2002; *Kuo et al.*, 2004], is to assume that all errors are uncorrelated. In this case (2) simplifies to

where *a* is the impact parameter and σ_{guess} and σ_{obs} are the standard deviations of the first-guess bending angle errors and the observed bending angle errors, respectively. It is worth noting that (2) also simplifies to (4) if both the observation and background error variances are constant with height and the two correlation functions defining **B**_{obs} and **B**_{guess} are identical except for a scaling factor so that **B**_{obs} ∝ **B**_{guess}.

[6] When applying this method, the observation errors are normally estimated dynamically from the upper part of the occultation and the standard deviation of the first-guess errors is assumed to correspond to a fixed fraction of the first-guess bending angle, typically 10% to 20%, as originally suggested by *Sokolovskiy and Hunt* [1996]. The popularity of this approach is mainly due to its simplicity and the fact that this technique has been demonstrated to work well in a number of studies (see the references above). In the following we will refer to this approach as the standard method. However, this method is not optimal mainly for two reasons. First, it is assumed that the first-guess error standard deviation corresponds to a fixed ratio of the first-guess bending angle though the accuracy of climate models vary with both season and geographical location. Second, this approach does not account for vertical correlation of the observation and first-guess errors.

[7] *Healy* [2001] applied the full matrix approach given by (1) assuming a Gaussian error correlation function with a correlation length of 6 km for the first-guess errors and uncorrelated observation errors. A fixed relative first-guess error of 20% and a fixed observation error standard deviation of 5 μrad were also assumed. *Gobiet and Kirchengast* [2004] also applied the full matrix approach, but in their study the observation error variance was estimated for each occultation from the upper part of the occultation following *Sokolovskiy and Hunt* [1996] and the relative error of the first guess was set to a fixed value of 15%. The error covariance for both the first-guess errors and the observations errors were assumed to be exponential with correlation lengths of 6 km and 1 km, respectively. To reduce biases between the climatology and the observations, *Gobiet and Kirchengast* [2004] selected their first guess from an independent search library of climate profiles.

[8] Though the full matrix approach in principle accounts for vertical error correlations, the efficiency is limited by the accuracy of the assumed covariance functions. Currently, only crude estimates are available for climate model error covariances and, moreover, the model error covariances are a function of season and geographical location as mentioned above. For that reason, application of the full matrix solution may not necessarily lead to any improvements compared to the standard method.

[9] *Gorbunov* [2002] presented a combined algorithm of ionospheric calibration and noise reduction based on SO. In that study, dynamic estimates of the magnitude of both observation and first-guess errors were applied, while vertical correlations were neglected. *Gorbunov* [2002] also suggested to reduce systematic deviations between climatology and observations by scaling the climatology on the basis of a best fit between observations and climatology in the height range of 40–60 km. This technique was also used by *Gobiet and Kirchengast* [2004]. By estimating errors dynamically for each occultation, geographical and seasonal variations are automatically accounted for. However, when dynamic estimation of both first-guess errors and observation errors is applied, special care should be taken if vertical correlations are neglected and refractivities are computed through the Abel transform. The reason for this is that the Abel transform is a low-pass filter. Hence observation errors and first-guess errors are only damped (or amplified) equally if the corresponding error covariances have the same shape. Generally, climate model errors have more low spatial frequency components than observation errors. It is therefore expected that the observation errors will be damped more by the Abel transform than the model errors. Thus the combination of the two profiles, given by (4), which results in the smallest errors in the bending angle profile, is not necessarily identical to the combination of the two profiles which results in the smallest errors in the corresponding refractivity profile. Consequently, if (4) is applied together with dynamic error estimation, too much weight could be given to the model profile. In that case, even if the error standard deviations have been estimated perfectly, the retrieved refractivity profile may well deviate more from the “true” profile than a corresponding profile computed through the standard method.

[10] In this study we present an approach where first-guess error covariances and observation error covariances of the ionospherically corrected bending angle profile are estimated dynamically. The combined profile is computed from (4) with modified standard deviations based on the estimated error correlation lengths.

[11] This study is organized in the following way: an analysis of how noise is damped by the Abel transform is given in section 2. In section 3 it is demonstrated how first-guess and observation error covariances can be estimated dynamically. Section 4 presents a comparison between the standard SO method and the technique presented in this study. Both methods are applied to one month of Challenging Minisatellite Payload (CHAMP) radio occultations, and the retrieved refractivity profiles are compared to corresponding profiles computed from the European Centre for Medium-Range Forecasts (ECMWF) analysis. The importance of the first-guess profile is also demonstrated by applying two different types of first-guess profiles derived from the same climate model but adjusted differently to the observations.